Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar

For a δ-function-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality d. It is shown that anomalous scaling appears in the first order of 1/d perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for its derivatives, in particular, for the dissipation field.